HINT: <no title>
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Start with the fact that the interior angles of a triangle sum to 180°.
STEP: Use the sum of angles in a triangle to find x
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We need to find two different unknown angles in the
diagram. We can find them using facts we know about triangles.
One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.
Interior angles of a triangle102,12°+x+38,94°=180°=180°
xx=180°−102,12°−38,94°=38,94°
You can also solve for x
using the fact that this triangle is isosceles (it has two equal
sides). Isosceles triangles have two congruent (equal) angles. In this
case the equal angles are ∠B and ∠C. And that means that x is equal to 38,94°.
Angle x has a measure of 38,94°.
STEP: Use the exterior angle theorem to find y
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The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.
The exterior angle theorem for
triangles tells us that an exterior angle is equal to the sum of the two
interior angles opposite the exterior angle. In this triangle, the
exterior angle DÂ C is equal to the sum of the opposite interior angles, ∠B and ∠C. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.
We can write an equation based on the exterior angle theorem to find the value of y:
Exterior angle of a triangle = The sum of the opposite interior angles
yyy=∠B+∠C=38,94°+38,94°=77,88°
You can also find y using supplementary angles. The two angles at A must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=77,88°.
The correct answers are:
- x= 38,94°
- y= 77,88°
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